Hi Allgonquin, BassLakeDan,
Thanks for your input.
I guess there’s another way to think about it:
In desert areas, there is a ton of airborne dust, which is basically very fine sand. Sand is relatively heavy, but still gets airborne. Apparently, sand can have a specific gravity from about 2.6 to 7.0 (give or take depending on purity and porosity). Steel definitely is denser, with a specific gravity around 7.8.
http://www.reade.com/Particle_Briefings/spec_gra.html
You might think,”Oh, well, steel is several times denser than sand, so it won’t be airborne.” However, one should return to scaling laws from undergraduate physics. Since steel is 3x times denser than silica, it can get airborne so long as the particles are 3x smaller. Below, I briefly describe the physics for this.
For a particle, the two main forces we’re interested in are the force due to air resistance (ie: small air currents), and the particles weight. When the force due to air resistance is at least as big as the weight of the particle, then it can easily take-off and get airborne. Very roughly speaking, the air-resistance of a small particle is proportional to it’s cross sectional area. But it’s weight is proportional to it’s volume. Let’s assume the particle is roughtly spherical.
Let c1 and c2 be constants (discussed below).
Let r be the radius of the spherical particle.
Force on the particle due to the small air currents in a typical room:
F_air = c1*r^2
Force of gravity due to the weight of the particle.
F_grav= c2*r^3
I’m cheating a little, because the force due to air, F_air, will depend upon the velocity of the air. (ie: the “constant” c1 will depend on air velocity.) But suppose we fix the velocity of the air to be something small, but typical of air-currents in a typical room (say with air conditioning or something; maybe several inches per second (I’m guessing based on watching steam or smoke in the kitchen)).
For gravity, we’ll adsorb various constants like the density of metal into the constant c2. This also includes things like the constant factor for the volume of a sphere, namely (4/3)*pi.
Now we can ask, what happens when the radius, r, gets very small? As r shrinkgs, the cube, r^3, goes to zero faster than the square, r^2. So at some point, r will be small enough so that F_air > F_grav. At that point, we have liftoff for our tiny metal particles.
Particle can fly when F_air >= F_grav which is the same as saying F_air/F_grav >= 1.0
F_air/F_grav = (c1/c2)*(r^2/r^3) = (c1/c2)/r = c3/r
where the constant c3 = (c1/c2).
So whatever the constant c3 is, if we make r small enough, then c3/r will be huge, in other words bigger than 1.0. For a fixed constant c3, we can always make r small enough so that:
c3/r >= 1.0 which means F_air >= F_grav.
Here is the key point and conclusion:
We have just shown that for a very simple model (spherical particles), you can always get airborne, so long as the particle is small enough.
In physics we give this type of r^2 versus r^3 analysis a name. We call it “surface-area versus volume” or “surface-area to volume ratio”, and it occurs all the time. What the math shows, is that as we shrink the particle size, it’s air resistance shrinks, but its weight shrinks even faster! So at some point, air-resistance beats particle-weight, and the particle can go airborne.
In practice, how small does r need to be for steel particles to float around a room with currents from air-conditioning? I don’t know, because I have not finished the calculation which would involve getting actual numbers for the constants c1 and c2.
But just consider that r is very very small…. With our super-fine grits these days, our grit-sizes are already micron or sub-micron sized. It is not hard to imagine that the swarf we generate could be about as big, or smaller than the grit size. So we’re talking about micron and sub-micron sized metal particles. Given that powdered glass (ie: very fine sand dust) can easily blow around in the wind, it’s not hard to imagine that steel could do the same, so long as it was more finely ground.
Our metal particles are smaller than a human red blood cell (6-8 microns)!
https://en.wikipedia.org/wiki/Red_blood_cell_distribution_width
Our analysis above shows that the critical particle size is inversely proportional to density (the density got adsorbed into the constant factor c2 as a linear factor). So, steel with a specific gravity if about 7.8, and sand with a specific gravity if around 2.6, gives us a density ratio of 7.8/2.6 = 3.0. So if sand-dust blows around with a particle size of r_sand, then we know that if the particle size for steel is just 3 times smaller, it can also blow around. If you believe sand-dust (or dirt-dust) can blow around in the air, then steel-dust can too, so long as the steel dust is about 3x finer.
If airborne particles of sand can have a radius of r_sand, then particles of steel of radius r_steel can blow around where
r_steel <= r_sand/3.0
The above is a very _approximate_ analysis. But this analysis of "area versus volume" is an example of the back-of-the-envelope calculations that physicists do all the time. And it also shows you something about how they try to reason and get rough intuition about things in the real world.
Here's a webpage about airborne particles that can be hazardous to people (scroll down to the section "Airborne Particles").
http://www.engineeringtoolbox.com/particle-sizes-d_934.html
Sincerely,
–Lagrangian
P.S. For you physics majors: For a proper analysis, one would need to compute the "settling time" for particles in air. This would be based on a more detailed analysis, including the terminal velocity of the particles. As you can imagine, the terminal velocity of smaller particles is much smaller. Once the settling time gets too long (hours, days, weeks, years…) then the particles are basically airborne. And if the terminal velocity is smaller than the typical microscopic fluctuations in air currents (ie: Brownian motion), then the particles will remain airborne indefinitely.
P.P.S. I wrote this post when I was very sleepy, so my apologies if it's not very well written. If you're wondering, I very much enjoyed being an undergraduate physics major.
P.P.S.S. A similar discussion about "surface-area versus volume" came up in a discussion about how to remove metal particles from the surface of a magnet. (The idea was to use a magnet to capture metal swarf from the sharpening.) You can read about that in this other post in the WickedEdge forums:
http://www.wickededgeusa.com/index.php?option=com_kunena&func=view&catid=6&id=2390&limit=6&limitstart=18&Itemid=63#2975
This kind of "surface-area versus volume" analysis occurs everywhere. In chemical reactions (reactions between two things often occurs on a surface where two chemicals meet (ie: burning wood), but the amount of chemical is based on volume). Also in energy analysis (heat/energy is exchanged on a surface, but the amount of energy/heat is often based on volume. The "surface-area versus volume" analysis is even mentioned early on in this technical book on the science of cutting (I forget if it's in the preface or the first chapter). I just started reading, so I'm only on Chapter 2.
_The Science and Engineering of Cutting_
Tony Atkins (2009)
http://www.amazon.com/The-Science-Engineering-Cutting-Biomaterials/dp/075068531X/ref=sr_1_1?ie=UTF8&qid=1337352735&sr=8-1
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